Symmetry group of a regular hexagon

The symmetry group of a regular hexagon consists of six rotations and six reflections. The six reflections consist of three reflections along the axes between vertices, and three reflections along the axes between edges.As with all groups, the composition of two or more symmetries is itself one of the twelve symmetries. Here we see a table of hexagonal rotation and reflection compositions, with the row operations applied before the column operations.

In this image we see the symmetry group D6 of a regular hexagon. The hexagon can be rotated six ways, and reflected six ways. Note that any combination of two or more of these operations will still result in one of these twelve configurations. While rotations are commutative in two dimensions, reflections are not, so any composition of symmetries involving reflection is dependent on the order in which they are applied. In the table above, the row operation is applied before the column operation.